2 Answers
2

Yes, this is right. For a correct Big $O$ estimate, expand out , find the dominant term and then simplify; don't take big $O$ estimates in intermediate steps. In your expression,the dominant term is $111n! n^3$ as you say which is $O(n^{3+n})$. You could even say your expression is $O(n!n^3)$ which is somewhat simpler and tighter.